In this article, Lucas maintains the falseness of Mechanism - the attempt to explain minds as machines - by means of Incompleteness Theorem of Gödel. Gödel’s theorem shows that in any system consistent and adequate for simple arithmetic there are formulae which cannot be proved in the system but that human minds can recognize as true; Lucas points out in his turn that Gödel’s theorem applies to machines because a machine is the concrete instantiation of a formal system: therefore, for (...) every machine consistent and able of doing simple arithmetic, there is a formula that it can’t produce as true but that we can see to be true, and so human minds and machines have to be different. Lucas considers as well in this article some possible objections to his argument: for any Gödelian formula we could, for instance, construct a machine able to produce it or we could put the Gödelian formulae that we had proved as axioms of a further machine. However - as Lucas underlines - for every of such machines we could again formulate another Gödelian formula, the Gödelian formula of these machines, that they are not able to proof but that we can recognize as true. More general arguments, such as the possibility to escape Gödelian argument by suggesting that Gödel’s theorem applies to consistent systems while we could be inconsistent ones, are moreover refuted by Lucas by maintaining that our inconsistency corresponds to occasional malfunctioning of a machine and not to his normal inconsistency; indeed, a inconsistent machine is characterized by producing any statement, on the contrary human being are selective and not disposed to assert anything. (shrink)

Responsibility is a key concept in our moral, social, and political thinking, but it is not itself properly understood. J.R. Lucas here presents a lively, broad, and accessible discussion of responsibility in various areas of human life, from personal and sexual relations to politics.

The legend of the encounter between Wilberforce and Huxley is well established. Almost every scientist knows, and every viewer of the BBC's recent programme on Darwin was shown,* how Samuel Wilberforce, bishop of Oxford, attempted to pour scorn on Darwin's Origin of Species at a meeting of the British Association in Oxford on 30 June 1860, and had the tables turned on him by T. H. Huxley. In this memorable encounter Huxley's simple scientific sincerity humbled the prelatical insolence and clerical (...) obscurantism of Soapy Sam; the pretension of the Church to dictate to scientists the conclusions they were allowed to reach were, for good and all, decisively defeated; the autonomy of science was established in Britain and the Western world; the claim of plain unvarnished truth on men's allegiance was vindicated, however unwelcome its implications for human vanity might be; and the flood tide of Victorian faith in all its fulsomeness was turned to an ebb, which has continued to our present day and will only end when religion and superstition have been finally eliminated from the minds of all enlightened men. Even churchmen concede that it was a disastrous defeat.1 Only Owen Chadwick strikes a note of caution, observing that the account given of the incident in Wilberforce's biography seems hardly consistent with an overwhelming defeat, and maintaining that the received account must be a largely legendary creation of a later date. (shrink)

That space and time should be integrated into a single entity, spacetime, is the great insight of Einstein's special theory of relativity, and leads us to regard spacetime as a fundamental context in which to make sense of the world around us. But it is not the only one. Causality is equally important and at least as far as the special theory goes, it cannot be subsumed under a fundamentally geometrical form of explanation. In fact, the agent of propagation of (...) causal influence is electromagnetic radiation. In this examination, the authors find support for a rationalist approach to physics, never neglecting experimentation, but rejecting a simple empiricist or positivist view of science. (shrink)

The issue is obscured by the fact that the word `space' can be used in four different ways. It can be used, first, as a term of pure mathematics, as when mathematicians talk of an `n-dimensional phase-space', an `n-dimensional vector-space', a `three-dimensional projective space' or a `twodimensional Riemannian space'. In this sense the word `space' means the totality of the abstract entities-the `points'-implicitly defined by the axioms. There is no doubt that there exist, iii this sense, non-Euclidean spaces, because all (...) that is claimed by such an assertion is that sets of non-Euclidean axioms constitute possible implicit definitions of abstract entities, that is to say that some sets of non-Euclidean axioms are consistent. If Kant or any other philosopher had denied this, he would have been wrong; but Kant himself took care not to deny it, 2 and there is little reason to suppose that any philosopher concerned about space has been using the word in this, the pure mathematician's, sense. (shrink)

The Conceptual Roots of Mathematics is a comprehensive study of the foundation of mathematics. Lucas, one of the most distinguished Oxford scholars, covers a vast amount of ground in the philosophy of mathematics, showing us that it is actually at the heart of the study of epistemology and metaphysics.

Equality is the great political issue of our time. Liberty is forgotten: Fraternity never did engage our passions: the maintenance of Law and Order is at a discount: Natural Rights and Natural Justice are outmoded shibboleths. But Equality—there men have something to die for, kill for, agitate about, be miserable about. The demand for Equality obsesses all our political thought. We are not sure what it is—indeed, as I shall show later, we are necessarily not sure what it is—but we (...) are sure that whatever it is, we want it: and while we are prepared to look on frustration, injustice or violence with tolerance, as part of the natural order of things, we will work ourselves up into paroxysms of righteous indignation at the bare mention of Inequality. (shrink)

In this paper Lucas comes back to Gödelian argument against Mecanism to clarify some points. First of all, he explains his use of Gödel’s theorem instead of Turing’s theorem, showing how Gödel’ theorem, but not Turing’s theorem, raises questions concerning truth and reasoning that bear on the nature of mind and how Turing’s theorem suggests that there is something that cannot be done by any computers but not that it can be done by human minds. He considers moreover how Gödel’s (...) theorem can be interpreted as a sophisticated form of the Cretan paradox, posed by Epimenides, able to escape the viciously self-referential nature of the Cretan paradox, and how it can be used against Mechanism as a schema of disproof. Finally, Lucas suggests some answers to the most recurrent criticisms against his argument: criticisms about the implicit idealisation in the way he set up the context between mind and machine; questions concerning modality and finitude, issues of transfinite arithmetic; questions concerning the need of formalizing rational inference and some questions about consistency. (shrink)

The application of Gödel’s theorem to the problem of minds and machines is difficult. Paul Benacerraf makes the entirely valid ‘Duhemian’ point that the argument is not, and cannot be, a purely mathematical one, but needs some philosophical premisses to be able to yield any philosophical conclusions. Moreover, the philosophical premisses are of very different kinds. Some are concerned with what is essential to being a machine—these are typically intricate, but definite, easily formalised by the mathematician, but unintelligible to the (...) layman: others attempt to capture what is essential to being a mind, a person or a self—these are typically intuitive, but vague; resistant to exact definition by the logician, but, none the less, widely used and well understood. Gödel’s theorem itself, like many other truths, can be taken either way: it can be taken as a formal proof sequence yielding certain syntactical results about a certain class of formal systems, but it can also be taken as giving us a certain type or style of argument, which we can understand, and, once having got the hang of it, adapt and apply in innumerable different circumstances. In my dispute with the mechanist, I take Gödel’s argument both ways: I first take it as an argument which the mechanist, even according to his own mechanist principles, must accept as scoring some point against his favourite machine; and then I hope that the mechanist, as a man, will see that he can do better and that this sort of argument will always apply against any form of mechanism he espouses. (shrink)

MANY thinkers deny the possibility of businessmen having responsibilities or ethical obligations. A businessman has no alternative, in view of the competition of the market-place, to do anything other than buy at the cheapest and sell at the dearest price he can. In any case, it would be irrational-if, indeed, it were possible-not to do so. Admittedly, there is a framework of law within which he has to operate, but that is all, and so long as he keeps the law (...) he is free to maximise his profits without being constrained by any moral or social considerations, or any further sense of responsibility for what he does. (shrink)

Abstract Quantum mechanics has seemed to defy all attempts to construe it realistically, but antirealism, like the many?worlds hypothesis, is even more difficult to accept. In order to give a realist construal of quantum mechanics, we need first to distinguish the objective and rational aspect of reality from the paradigmatic thing?like aspects of having determinate physical properties: quantum?mechanical entities may be real in the former sense though not in the latter. Anti?realist arguments are based on the difficulty of giving an (...) account of quantum?mechanical collapse and the apparent superluminal velocities involved. Objections to superluminal velocities on the score of the special theory of relativity are found not to be conclusive, and the price?there being some preferred frame of reference?to be acceptable. A sketch of a probabilistic account of quantum?mechanical collapse is offered, which makes the difference between the macro? and the micro?world a matter of degree rather than kind. If that, or some other, account proved acceptable, we could be quantum?mechanical realists, though quantum?mechanical reality would be very different from that of material objects in hardware shops. (shrink)

There was once a leak from Hebdomadal Council. The Assessor told her husband, who told my wife, who told me that Monday afternoon had been spent discussing what Lucas would say if various courses of action were adopted, leading to the conclusion that it would be best to do nothing. I was flattered, but a bit surprised. The tide of philosophical scepticism had ebbed, and it was generally allowed that a reasonable way of discovering what someone would say was to (...) ask him. Dick Southwood did: he would quiz me in Common Room â€“ sometimes ending "Thank you for letting me bounce these ideas off you" â€“ and had reliable information about how one member of Congregation would react to various proposals. And not only me: he was a listening Vice-Chancellor, who used to bike from Wellington Square to Merton for lunch, greeting many as he passed them, and ready to stop if occasion warranted it. Of course, there are many other leaks. I remember once attending a meeting in the Town Hall to argue for cycle tracks, and someone coming up to me, and saying, "Youâ€™re having a tussle with Council, arenâ€™t you? I think you ought to see the minutes of their latest meeting"; the next day there was a copy in my pigeon hole, giving me just the ammunition I needed. What members of Congregation tend the forget is the existence â€“ the other side of the green baize door, so to speak â€“ of a corps of bedells. (shrink)

IT is ungenerous to pick holes in The Concept of Law. It is a great work. Its clarity is luminous, and its argument sustained and convincing. Hart is eminently successful in rescuing the concept of law from the Legal Realists, the Positivists, and the Formalists, who attempt to straitjacket it within schemata which are too narrow or too vague to give an adequate elucidation of it. But sometimes Hart is not carried along by his arguments as far as he should. (...) He makes too many concessions to his opponents, and his own account of the law is, in consequence, too formalist, in spite of having himself adduced cogent considerations elsewhere for rejecting the purely formalist line of argument. The rule of recognition, although important, is not fundamental. We should, rather, see law as a social phenomenon, to be distinguished from other social phenomena, but intelligible only in a social context, and not-as lawyers are too ready to suppose-an autonomous discipline which can be explained and understood entirely in its own terms. (shrink)

[D. H. Mellor] Kant's claim that our knowledge of time is transcendental in his sense, while false of time itself, is true of tenses, i.e. of the locations of events and other temporal entities in McTaggart's A series. This fact can easily, and I think only, be explained by taking time itself to be real but tenseless. /// [J. R. Lucas] Mellor's argument from Kant fails. The difficulties in his first Antinomy are due to topological confusions, not the tensed nature (...) of time. Nor are McTaggart' s difficulties due to the tensed nature of time. The ego-centricity of tensed discourse is an essential feature of communication between selves, each of whom refers himself as 'I', and is required for talking about time as well as experience and agency. Arguments based on the Special Theory are misconceived. Some rest on a confused notion of 'topological simultaneity'. In the General Theory a cosmic time is defined, as also in quantum mechanics, where a natural present is defined by a unique hyperplane of collapse into eigen-ness. (shrink)

It is meet and right that pride and humility should be the two human characteristics on which University sermons have to be preached. Left to myself, although I might have picked on my modesty as something I should share with you, I should have given the preeminence to other among my sins than pride. My greed, my sloth, my avarice or, in this salacious age my lust, are subjects on which I could tell you much that might interest you. Pride (...) lacks immediate appeal. We are not sure what it is, or whether it is a bad thing, when we think of it in purely individual terms. But when we consider it collectively, we can see that it is, together with humility, something Oxford is peculiarly well qualified to preach on. We all of us are proud of our university. We were proud, and our schoolmasters were proud, when we first got our places here. We are, dons and undergraduates alike, proud of our colleges, each grateful that good fortune has brought him to the best college in Oxford, and anxious that everyone else should secretly acknowledge it to be the best. Our parents were proud when we took our degrees, and although we profess to be unconcerned with classes, we are deeply content to record our firsts when occasion requires us to do so, or have our contemporaries allude to them as opportunity offers. We are studious, as dons, not to pull rank, safe in the knowledge that others will do it for us, and that we shall receive the deference due to a fellow of an Oxford college. In an age that is egalitarian in theory but elitist at heart, Oxford men have benefited greatly, as other forms of social eminence have been eroded, leaving a clear field for our own claims to public esteem, which are, if not entirely unchallenged, still generally allowed. Oxford is, as we like to be told by outsiders, a centre of excellence, and a lot of the resplendence rubs off on us, not altogether undeservedly. It is, as we corporately admit on Commemoration Sunday, largely due to our having entered into other men's labours.. (shrink)

The Unity of Science is often thought to be reductionist, but this is because we fail to distinguish questions from answers. The questions asked by different sciences are different---the biologist is interested in different topics from the physicist, and seeks different explanations---but the answers are not peculiar to each particular science, and can range over the whole of scientific knowledge. The biologist is interested in organisms--- concept unknown to physics---but explains physiological processes in terms of chemistry, not a mysterious vital (...) force. The replacement of Laplacian determinism by quantum mechanics further erodes the tendency towards reductionism. The answers given in different explanations are not subsumed under one complete theory; and quantum mechanics does not have a concept of haecceitas, "this-i-ness" which would make its entities the fundamental constituents of everything. (shrink)

South. So we have agreed to bury intuitionism. Well, I dare say it is right. But we ought to bury some of the grave-diggers too. Some of the things that Ross said are no doubt wrong, or at least misleading: but they are a lot less wrong than most of the things said since the war.

The ontological argument has run for a long time, regularly refuted, regularly re-appearing in a new form. Something can be learnt from its longevity. Its proponents must be on to something, or it would not have survived its many refutations. But equally, it must have been much misformulated, or it would not have seemed evidently fallacious to its many critics. Perhaps it does express a deep philosophical intimation. Certainly it has been taken to prove more than it really can establish. (...) Like many other philosophical arguments it has suffered by being made out to be more rigorous than in the nature of the case it can be. For some philosophers addressing some questions it may have been decisive in leading them to adopt one of the few options open to them: but it is quite inconclusive for others, with different presuppositions or different problems, and cannot be reduced into a valid proof cogent for all comers and compelling them to accept the conclusion it claims to demonstrate. (shrink)

“Towards a Theory of Taxation” is a proper theme for an Englishman to take when giving a paper in America. After all it was from the absence of such a theory that the United States derived its existence. The Colonists felt strongly that there should be no taxation without representation, and George III was unable to explain to them convincingly why they should contribute to the cost of their defense. Since that time, understanding has not advanced much. In Britain we (...) still maintain the fiction that taxes are a voluntary gift to the Crown, and taxing statutes are given the Royal Assent with the special formula, “La Reine remercie ses bons sujets, accepte leur benevolence, et ainsi le veult” instead of the simple “La Reine le veult,” and in the United States taxes have regularly been levied on residents of the District of Columbia who until recently had no representation in Congress, and by the State of New York on those who worked but did not reside in the State, and so did not have a vote. Taxes are regularly levied, in America as elsewhere, on those who have no say on whether they should be levied or how they should be spent. I am taxed by the Federal Government on my American earnings and by state governments on my American spending, but I should be hard put to it to make out that it was unjust. Florida is wondering whether to follow California in taxing multinational corporations on their world-wide earnings. (shrink)

Plato was the first feminist. In the Republic he puts forward the view that women are just the same as men, only not quite so good. It is a view which has often been expressed in recent years, and generates strong passions. Some of these have deep biological origins, which a philosopher can only hope to recognize and not to assuage. But much of the heat engendered is due to unnecessary friction between views which are certainly compatible and probably correct. (...) And here a philosopher can help. If we can divide the issues neatly, at the joints, then we need not quarrel with one another for saying something, probably true, because what is being maintained is misconstrued and taken to mean something else, probably false. (shrink)

Whatever good or ill it did to Guy Fawkes, his resuscitation at the hands of Bernard Williams has, by any utilitarian reckoning, been a Good Thing. A casual glance at the literature that has accumulated over the past thirty five years leaves no doubt that the topic has been reduplicated many times over, to the great enjoyment of undergraduates, who have been able to write science fiction under the guise of essays in the Philosophy of Mind, and of dons, who (...) in an age of cvs and Assessments, have been able to notch up page after page of counter-replies to replies to rebuttals of previous papers, not to mention an often welcome tally of references in the citation index. But the actual arguments adduced by Williams can be turned to support a much more traditional view of the self, as a necessarily unique agent whose individuality is established by his capacity for autonomous action. (shrink)

Equality in the present age has become an idol, in much the same way as property was in the age of Locke. Many people worship it, and think that it provides the key to the proper understanding of politics, and that on it alone can a genuinely just society be reconstructed. This is a mistake. Although, like property, it is a useful concept, and although, like property, there are occasions when we want to have it in practice, it is not (...) a fundamental concept any more than property is, nor can having it vouchsafe to us the good life. In an earlier paper I argued against equality by showing that the concept of equality was confused and that many of the arguments i egalitarians adduced were either invalid or else supported conclusions I which were not really egalitarian at all. Many egalitarians, however, have complained that my arguments were not fair, because I had failed to elucidate the concept adequately, or because the position I attacked was not one that any egalitarian really wished to maintain, or because I had overlooked other arguments which were effective in establishing egalitarian conclusions, or because the positive counter-arguments of my own I put forward more as a matter of taste than of serious political commitment. In this paper, therefore, I want to elucidate the concept more fully, concede what I should to my critics, point out that, even so, their conclusions do not follow, and give further reasons not only for supposing that egalitarian arguments are invalid but for discerning positive merits in some forms of inequality. (shrink)

Thus far the logic out of which mathematics has developed has been First-order Predicate Calculus with Identity, that is the logic of the sentential functors, ¬, →, ∧, ∨, etc., together with identity and the existential and universal quotifiers restricted to quotify- ing only over individuals, and not anything else, such as qualities or quotities themselves. Some philosophers—among them Quine— have held that this, First-order Logic, as it is often called, con- stitutes the whole of logic. But that is a (...) mistake. It leaves out Second-order Logic, which we need if we are to characterize the natural numbers precisely, and pays scant attention to the logic of relations, especially transitive relations, which is the key to much of modern mathematics. Quine’s argument for restricting logic to First-order Logice was based on the grounds that only First- order logical theories display “Law and Order” and himself regards modal logic as belonging with witchcraft and superstition.1 Pred- icates are ontologically more suspect than individuals, and have a different logic, which is liable to give rise to paradox and inconsis- tency. Moreover, Second-order Logic lacks the completeness that First-order Logice has, which provides a pleasing parallel between syntactic and semantic notions, and argues for the analyticity of deductive logic. (shrink)

My sights in this paper are trained on facts. Most people think that they know what facts are; that while their friends often, and themselves occasionally, are ignorant of the facts, at least they know what sort of things facts are---they can recognise a fact when they see it. Facts, in the popular philosophy of today, are good, simple souls; there is no guile in them, nor any room for subjective bias, and once we have made ourselves acquainted with them, (...) we have reached the beginning and summit of all wisdom. (shrink)

"Ich liebe dich 3" the swains in mountain valleys of Austria inscribe on their presents to those to whom they plight their troth. The pun is a rare one in German. Only in remote valleys does the word for `three' rhyme with joy; and the word for `true' is usually..

Elijah foretold evil for Ahab in the name of the Lord. ‘I will bring evil upon you; I will utterly sweep you away, and will cut off from Ahab every male, bond or free in Israel’ … but when he heard those words, he rent his clothes, and put sackcloth upon his flesh, and fasted and lay in the sackcloth, and went about dejectedly. And the word of the Lord came to Elijah saying ‘Have you seen how Ahab has humbled (...) himself before me? Because he has humbled himself before me, I will not bring the evil in his days, but in his son's days I will bring evil upon his house.’. (shrink)

Plato was politically incorrect---gloriously incorrect: hard to ignore and difficult to refute. Read An Engagement with Plato's Republic to argue with him or against him, for contemporary orthodoxies or against them. ``Plato was the first feminist. Women were the same as men, only not so good.''.

Whatever good or ill it did to Guy Fawkes, his resuscitation at the hands of Bernard Williams has, by any utilitarian reckoning, been a Good Thing. A casual glance at the literature that has accumulated over the past thirty-five years leaves no doubt that the topic has been reduplicated many times over, to the great enjoyment of undergraduates, who have been able to write science fiction under the guise of essays in the Philosophy of Mind, and of dons, who in (...) an age of cvs and Assessments, have been able to notch up page after page of counter-replies to replies to rebuttals of previous papers, not to mention an often welcome tally of references in the citation index. But the actual arguments adduced by Williams can be turned to support a much more traditional view of the self, as a necessarily unique agent whose individuality is established by his capacity for autonomous action. (shrink)